Best Known (19, 26, s)-Nets in Base 4
(19, 26, 514)-Net over F4 — Constructive and digital
Digital (19, 26, 514)-net over F4, using
- trace code for nets [i] based on digital (6, 13, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
(19, 26, 886)-Net over F4 — Digital
Digital (19, 26, 886)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(426, 886, F4, 7) (dual of [886, 860, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using
(19, 26, 63010)-Net in Base 4 — Upper bound on s
There is no (19, 26, 63011)-net in base 4, because
- 1 times m-reduction [i] would yield (19, 25, 63011)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1125 908201 826604 > 425 [i]